Here, theoretical uncertainties in the predictions of inner fission barrier heights in superheavy elements have been investigated in a systematic way for a set of state-of-the-art covariant energy density functionals which represent major classes of the functionals used in covariant density functional theory. They differ in basic model assumptions and fitting protocols. Both systematic and statistical uncertainties have been quantified where the former turn out to be larger. Systematic uncertainties are substantial in superheavy elements and their behavior as a function of proton and neutron numbers contains a large random component. The benchmarking of the functionals to the experimental data on fission barriers in the actinides allows to reduce the systematic theoretical uncertainties for the inner fission barriers of unknown superheavy elements. However, even then they on average increase on moving away from the region where benchmarking has been performed. In addition, a comparison with the results of non-relativistic approaches is performed in order to define full systematic theoretical uncertainties over the state-of-the-art models. Even for the models benchmarked in the actinides, the difference in the inner fission barrier height of some superheavy elements reaches $5-6$ MeV. This uncertainty in the fission barrier heights will translate into huge (many tensmore » of the orders of magnitude) uncertainties in the spontaneous fission half-lives.« less